Fibrewise injectivity lifts and leaves a flat cokernel over a Noetherian target

For $n\ge 1$ let $u_n:N\mathrm{/}{\mathfrak{m}}^{n}N\to M\mathrm{/}{\mathfrak{m}}^{n}M$. The hypothesis is that $u_1$ is injective. Suppose $u_n$ injective. The short sequence $0\to {\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}\to R\mathrm{/}{\mathfrak{m}}^{n+1}\to R\mathrm{/}{\mathfrak{m}}^n\to 0$ remains exact after tensoring with the $R$-flat module $M$. For $N$, tensoring gives the analogous sequence, right exact though not necessarily injective at its left. Both left tensor terms identify with the corresponding residue modules tensored over $k=R\mathrm{/}\mathfrak{m}$ with the vector space ${\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$. Since $u_1$ is injective, tensoring it over the field $k$ preserves injection on these left terms. A diagram chase with the two sequences and $u_n$ shows that $u_{n+1}$ is injective. Thus induction gives injectivity for every $n$.

If $z\in \ker u$, the image of $z$ under each $u_n$ is zero, so $z\in {\mathfrak{m}}^{n}N$ for every $n$. By [F2], their intersection is zero; hence $u$ is injective.

Let $I⊊R$ be any ideal. The map $R\mathrm{/}I\to S\mathrm{/}IS$ is local with Noetherian local target; $N\mathrm{/}IN$ is finite over that target, and $M\mathrm{/}IM$ is flat over $R\mathrm{/}I$ by [F1]. The reduction of $u_I:N\mathrm{/}IN\to M\mathrm{/}IM$ modulo the maximal ideal $\mathfrak{m}\mathrm{/}I$ is the original injective fibre map. Apply steps 1.1–2.1 to $u_I$ over this quotient local map: $u_I$ is injective. The case $I=R$ is trivial.

Put $C=\mathrm{coker}u$. Step 2.1 makes $0\to N\to M\to C\to 0$ exact, and step 3.1 makes $N\mathrm{/}IN\to M\mathrm{/}IM$ injective for every ideal $I$. By [F3], ${\mathrm{Tor}}_1^R(R\mathrm{/}I,C)=0$ for every $I$, so $C$ is $R$-flat. The Axiom of Choice is inherited at the cited Krull-intersection and flatness boundaries.